Precision measurement of atom-dimer interaction in a uniform planar Bose gas

Cold quantum gases, when acted upon by electromagnetic fields, can give rise to samples where isolated atoms coexist with dimers or trimers, which raises the question of the interactions between these various constituents. Here we perform microwave photoassociation in a degenerate gas of ${}^{87}\mathrm{Rb}$ atoms to create weakly bound dimers in their electronic ground level. We determine the complete energy diagram of one hyperfine manifold of the least-bound level, which we accurately reproduce with a simple model. Then, from the density-induced shift of the photoassociation line, we measure the atom-dimer scattering length for two weakly bound states of the molecular potential.

Figure 1

(a),(b) Clouds used for microwave spectroscopy with either (a) a single component state $|f=1,m_f=0,\pm 1\rangle$ or (b) a binary mixture of these hyperfine states (bar length: $10\mu \mathrm{m}$). (c) Relevant levels for a pair of atoms occupying the $f=1$ or $f=2$ hyperfine sublevels of the electronic ground state of ${}^{87}\mathrm{Rb}$. The dissociation limits of molecular state manifolds, represented as dashed lines, are separated by $h{\nu }_{\mathrm{hf}}\approx h\times 6.8\mathrm{GHz}$. The molecular potentials are represented by thick continuous lines. The graph is limited to relatively large interatomic distances where van der Waals interaction dominates, hence the superposition of singlet and triplet potentials at the scale of the figure. We also indicate the values of the total spin angular momentum $F$ of the dimer. Here, we focus on the least-bound vibrational levels $n=-1$ and $n=-2$ of the $(f=1;f=2)$ subspace, with zero orbital angular momentum. These levels are located $\sim 25$ and 642 MHz below the dissociation energy, respectively. The dimers are produced by microwave photoassociation of two atoms either both in $f=1$ (${\mathrm{M}}_1$ and ${\mathrm{M}}_2$ lines) or both in $f=2$ (${\mathrm{M}}_1^{\prime }$ line).

Figure 2

Energy diagram of the $(f=1;f=2)$ subspace. The colors of the experimental points encode the composition of the initial atomic state. The set of solid lines is the result of the simple model described in the text, with two adjustable parameters $U$ and $E_0$. The state $|{\mathrm{\Psi }}_0^{(n=-1)}\rangle$ used for measuring the atom-dimer scattering length is highlighted with a red solid line. The numbers on the right give, for each state, the quantum number $M_F$ associated with the $z$ component of the spin angular momentum $\widehat{\mathit{F}}$.

Figure 3

Frequency shift of the ${\mathrm{M}}_1$ line as a function of the 2D density of the gas. The three different symbols correspond to a measured relative depletion signal of 20% (square), 14% (diamond), and 8% (circle). All data are adjusted by a common linear fit. The left-hand inset shows the result of a similar measurement for the second-to-least bound level (${\mathrm{M}}_2$ line in Fig. 1). The right-hand inset shows a typical microwave photoassociation signal for $n_a=95$ $\mu {\mathrm{m}}^{-2}$. The variation of the width of the photoassociation signal with density is reported in [53].